The Dimensions of the Parthenon

1. Vitruvius, in listing the writers on architecture that constitute the sources
of his sketchy compendium, mentions that the architect of the Parthenon, Ictinos,
wrote, together with a certain Carpion, a book in which he explained the proportions
of this temple.1 This piece of
information makes us bemoan that all the ancient literature of this type is
lost, except for the few rules about proportions that Vitruvius culled from
it. Nevertheless, we may console ourselves with the thought that since Ictinos
was an architect, his main form of communication was through the stones of the
Parthenon and not through the pen. It may be surmised that Ictinos collaborated
with the otherwise unknown Carpion, just because writing was not his trade.
But one fact is clear: since Ictinos wrote a book on the proportions of the
Parthenon, it follows that these proportions constituted a system that was per
se of intellectual significance and appeal. The architecture of the Parthenon
cannot have been a hit-and-miss affair as claimed by Hill, Dinsmoor and others.

Since the book of Ictinos is lost, we must let him speak to us through the
stones of the Parthenon. These stones were carefully put together so that they
would last for eternity, in the sense that at least they should be aere perennius.
The Parthenon was transformed into a Christian church and later was used as
a fortress from which artillery fired and which was fired at for centuries.
Bombs still fell on the Parthenon during the Greek War of Independence in the
first part of the last century. The most substantial damage was that caused
by the explotion of the Turkish gunpowder deposit when Morosini’s gunners scored
a bull’s eye on September 26, 1678. There was also another explosion of the
gunpowder deposit at an earlier date which cannot be established with the same
exactness. In spite of this, a goodly portion of the stones is still there to
convey Ictinos’ message. And his message is a construction so tightly knit intellectually
and physically that it can still be understood today, if one is willing to listen
to it. The greatest wrong against Ictinos’ work was not committed by Christians,
Turks, Venetians, and so on, but by the scholars who, because of laziness and
conscious intellectual obscurantism, did not want to follow him in his extreme
concern with exact measurements and proportions.

2. Vitruvius expains that the peripteros of temples was no truly a part of
the sacred architecture. The sacred part was the cella, which the Greeks called
sekos, ”sacred enclosure” whereas the peripteros was added inorder to give greater
relief to the structure and to provide a shelter for the crowds in case of rain.
Hence, the existence of the peripteros is related to the circumstance that sudden
and brief showers are characteristic of the Mediterranean climate.

That the peripteros is an additional element was particularly clear in the
case of the Old Temple, since this colonnade was added in a second moment. This
fact was taken into account in planning the Parthenon, which was specifically
intended to make up for the Persian destruction of the Old Temple. Hence, the
architecture of the Parthenon can be better understood if the proportions of
the cella are considered first.

The cella of the Parthenon was an amphiprostyle temple, as was the cella of
the Old Temple. The autonomy of the cella of the Parthenon was emphasized by
placing it on two steps, of which the higher represented the stylobate and the
lower represented the euthynteria, ”the levelling course.”

It was decided to make the cella of the new temple as wide as the total Old
Temple, which had a width of 70 geographic feet = 77.77 trimmed lesser feet.
But, since the width of the cella had to be divided into intercolumnia measured
in round numbers of trimmed lesser feet, the width came to a fraction of a foot
more than 77.77.

Since the total Old Temple had 6 columns on the fronts, the cella of the Parthenon
was provided with porches of 6 columns at each of the two ends. Balanos calculated
the spacing of the columns of the cella by measuring the blocks of the stylobate
one by one and assuming that the columns stood exactly on the junctions of the
blocks. He reports that the spacing of the columns, expressed in mm., is the
following:

Western colonnade

899

3661

4197

4185

4185

3685

899

Eastern colonnade

899

3667

4207

4171

4187

The north end of the eastern colonnade is too severely damaged to permit a
measurement. The blocks form a curve, but their distances were measured by Balanos
projected on the line joining the two ends. The method used by Balanos was such
that the total result is inferior to the real one; I shall examine Balanos’
method in detail in dealing with the columnation of the peripteros. Here it
may be enough to observe that Balanos’ figures for the western colonnade of
the peripteros add up to 21,711 mm., which is certainly inferior to reality.

In spite of the fact that Balanos’ figures for the intercolumnia deviate strongly
from each other, one can easily infer from them what was the norm for the columnation:

Normal intercolumnium

15 7/48 feet

4,202.2 mm.

Corner spaces

16½ feet

4,577.9 mm.

The latter were divided as follows:

Space between the axis of normal column and the axis
of corner column

13¼ feet

3,676.2 mm.

Space between the axis of corner column and the edge
of the stylobate

3¼ feet

901.7 mm.

Hence, the width of the cella was computed as follows:

3 normal intercolumnia of

15 7/48 feet

452 1/48 feet

2 corner spaces of

16½ feet

33 feet

Total width

78 21/48 feet

21,762.4 mm.

This is the length measured on a curved line. Since all the horizontal lines
of the Parthenon were curved upwards and inwards, the width measured on a straight
line was something less. The contraction due to the curvature proves to have
been 6/48 or 1/8 of foot, that is, 34.7 mm. The cella measured in a straight
line had a width of 78 15/48 feet or 21,727.7 mm.

Later I will explain why the cella was made larger at the western end by 1/48
of foot, so that there its width was 78 1/3 feet or 21,733.5
mm. Balanos, who measured the width of the temple in correspondence with the
pedestal of the statue of the Goddess, which is at about the middle point of
the cella, found a width of 21,731 mm.

3. According to Vitruvius, the space between the columns of the peripteros
and the side of the cella should have the width of an intercolumnium.2
This rule was followed in the Parthenon in the sense that the peripteral spaces
on the flanks were made equal to the spaces assigned to the corner columns of
the cella, that is, 16½ feet of 4,577.9 mm. According to Balanos the interval
between the stylobate of the peripteros and the stylobate of the cella averages
4,578 mm. on the north side and 4,584 mm. on the south side. The difference,
which according to Balanos is 6 mm., results from the fact that 1/48 of foot,
or 5.78 mm., was added to the south side in order to obtain a width of 111 1/3
feet for the eastern front of the temple.

In this calculation of the peripteral space it is implied that the front of
the peripteros should have one column more at each side than the cella. The
cella has fronts of 6 columns and the peripteros has fronts of 8 columns.

By theoretical reckoning one arrives at the conclusion that the total temple
had the following width:

Width of the cella

78 15/48 feet

Width of the peripteros

33 feet

Total width

111 15/48 feet

But 1/48 of foot was added tot he south side, making the total
width 111 1/3 feet or 30,889.3 mm. This is the width measured
on a straight line.

The width on a curved line was reckoned by adding 1/48 of foot for each peripteral
space. Hence the width on a curved line was reckoned as follows:

Width of the cella

78 21/48 feet

21,762.4 mm.

The peripteral space

33 2/48 feet

9,167.4 mm.

Addition on the south side

1/48 feet

5.8 mm.

Total width of the temple

111½ feet

30,935.6 mm.

In the cella, which has a front of 6 columns, the contraction due to the curvature
is 6/48 or 1/8 of foot; in the peripteros, which has a front with 8 columns,
the contraction is 8/48 or 1/6 of foot.

4. Stuart and Revett had concluded that the Parthenon was called hekatompedos
because the front of the stylobate measured 100 feet, the kind of feet that
Pliny describes as being 25/24 of the Roman foot. Stuart and Revett were correct
in their first approach to the dimensions of the Parthenon, but they did not
have access to the additional information that permits us to see the problem
as much more complex.

The temple was planned by trimmed lesser feet. This had been the unit employed
in constructing the older and most sacred parts of the Old Temple which the
Parthenon was intended to replace. This unit had also been employed in planning
Parthenon II, on top of which the Parthenon was erected. But the last addition
to the Old Temple, the peripteral colonnade, was planned by geographic feet
(stylobate of 70 x 140 feet). Hence, the stylobate of the peripteral colonnade
of the Parthenon, too, was planned in geographic feet, 100 x 225.

The rest of the Parthenon was planned by trimmed lesser feet, but the combination
of two different modules of foot did not cause any difficulty, since the two
units relate as 9:10. The geographic foot is 25/24 of the Roman foot, and the
trimmed lesser foot is 15/16 of the Roman foot: 16/15 x 25/24 = 10/9. Reckoning
in trimmed lesser feet, the stylobate of the Parthenon was planned as 111.111
(=111 1/9) x 250 feet.3

A stylobate of 100 x 225 geographic feet or 111 1/9 trimmed
lesser feet was merely a starting point for the architects, because they had
to divide the sides of the stylobate into so many intercolumnia expressed in
round figures. Eight columns were planned for the fronts and seventeen for the
sides, counting the corner columns twice. As usual in Doric temples, the corner
intercolumnia had to be narrower, the narrowness being compensated by making
the corner columns thicker. The origin of this practice seems to be the following:
if the sides of the stylobate were divided exacly into so many identical intercolumnia,
the axes of the corner columns would fall on the corners of the stylobate; hence,
the corner columns were displaced to the inside.

For the normal intercolumnia the architects chose the length of 15½ trimmed
lesser feet, or 4,300.5 mm. Since the columns, which were measured on a different
module of foot, came to about 7 trimmed lesser feet in their maximum dimension,
the architects apparently reckoned the intercolumnium as 2 columns plus 1½.
For the corners they added 1½ feet more, arrovomg at 17 feet or 4,716.6
mm. for the distance between the axis of the last normal column and the edge
of the stylobate. This space of 17 feet was divided as follows: 13 feet, 5 fingers
or 3,693.5 mm for the distnace between the axis of the last normal column and
the axis of the corner column, and 3 feet, 11 fingers, or 1,023.1 mm. between
this axis and the edge of the stylobate.

For the fronts the architects reckoned:

5 basic intercolumnia of

15½ feet

77½ feet

2 corner space of

17 feet

34 feet

Total length of fronts

111½ feet, or 21,762.4 mm.

For the flanks they reckoned:

14 basic intercolumnia of

15½ feet

217 feet

2 corner space of

17 feet

34 feet

Total length of flanks

251 feet, or 69,639.7 mm.

In other words, they added ½ foot to the front and 1 foot to the flanks.
This explains why the temple, although called hekatompedos, had a width that
was slightly more than 100 geographic feet and why the length is a trifle more
than 9/4 of the width, as Stuart and Revett had noticed.

5. The reason for which the Society of Dilettanti in 1846 sent Penrose to measure
the Parthenon was to test the theory of John Pennethorne that what appears as
straight and parallel in Greek architecture of the best period is generally
curved or inclined, because this is the only way to obtain the optical effect
of a straight line. Immediately upon his return to England in 1847 Penrose published,
as the first result of his survey, a paper entitled ”Anomalies in the Construction
of the Parthenon,” in which he proved that the lines of the stylobate of the
Parthenon are curved to the inside, as Pennethorne had maintained on the basis
of the study of human optics. In turn, when Pennethorne in 1878 brought out
his Geometry and Optics in Ancient Architecture, he made abundant use
of Penrose’s data about the curvature of the stylobate of the Parthenon. In
spite of his great concern with this curvature, Penrose never raised the question
of how it affected the calculation of the length of the sides and of the intercolumnia.
Probably he left this basic issue in the dark because the problem of metrology
of the stylobate remained a mystery to him, once he had rejected the metrological
interpretation of Stuart and Revett.

The calculations of the length of the intercolumnia that I have presented apply
to the stylobate seen as curved lines. If the stylobate is seen as a rectangle,
the intercolumnia and the total length of the sides will be slightly less than
what I have calculated, particularly on the flanks. It is regrettable that Penrose
did not realize that he should have measured the intercolumnia, first, directly
from axis to axis and, second, according to the projection of this interval
on the straight line joining the angles of the stylobate. He measured the stylobate
on a straight line and measured the intercolumnia on a curve, without relating
the two.

All the Herculean labor that has been spent for the purpose of establishing
to the millimeter the precise size of the stylobate has been misdirected. My
conclusion is that the stylobate, measured on the curved lines, had been planned
as

30,935.6 x 69,639.7 mm.

Penrose obtained the following results on the two fronts:

East side

30,889.7 mm

West side

30,896.4 mm

Difference

6.7 mm

Penrose was dismayed by this difference, since he wanted to prove, against
the prevailing opinion of his colleagues, that the temple had been planned and
erected with great accuracy and precision of measurement. He feared that they
would grasp at any straw in order to argue that the temple had been built helter-skelter.
And, indeed, this fear was perfectly justified. For instance, the highly authoritative
Topographie von Athen by Walther Judeich declares, apparently on the
asis of a distorted reading of Magne’s report, that the cella measured in m.
”59.02; 22.34; or rather [beziehungsweise] 59.83; 21.72.”4
In the widely adopted textbook on Greek architecture by Robertson the same figures
are cited to the effect that the cella is 21.72 mm. on the east side, 22.34
on the west side, 59.02 on the north side, and 59.83 on the south side. It is
true that the explosion of 1687 caused serious deformations of the southern
part of the cella, but not in the range of the figures just mentioned. In order
to forestall this kind of perversions, Penrose tried to argue that the apparent
small irregularities in the dimensions of the stylobate of the peripteros are
due to displacements of the blocks from their original positions.

The figures originally obtained by Penrose are practically perfect. By theoretical
calculations | have arrived at a width of 30,889.3 mm for the eastern front
and of 30,895.1 mm (1/48 of foot more) for the western front. Balanos, who measured
the meridian section in correspondence with the pedestal of the statue, reports
a width of 30,893 mm. He obtained a figure intermediary between the two just
mentioned, as one would expect.5

Penrose tried to account for the small excess on the western side by assuming
that the blocks had come apart at the joints. Nevertheless, he himself reported
that the blocks appear perfectly joined. He also pointed out that the blocks
have an upward curvature so that they from an arch, with the result that the
very weight of the blocks plus the weight of the columns pressing on them would
keep them in place. A lengthening of the sides oculd have been caused only by
displacement of the corners, but there is evidence that the corners did not
move, since the upwards curvature of the sides has remained symmetric and regular.
This was ascertained by Penrose and confirmed by Balanos. What has changed is
the inward curvature of the blocks. The explosion has almost obliterated this
curvature on all sides except the north side—so much that Penrose completely
neglected this horizontal curvature, concentrating on the vertical curvature.
But Balanos reports that the north side has a symmetric horizontal curvature
that reaches a maximum of 160 mm. He reports that the damage caused by the explosion
has almost cancelled this curvature on the other three sides, but one can infer
the extent of this curvature by studying the inward curvature indicated by the
capitals and higher structures. Unfortunately, in spite of the sums contributed
by governments and learned societies for the study of the architectural structure
of the Parthenon, no scientific study of the impact of the explosion has ever
been attempted.

Penrose, even though he recognized that there was no evidence for a shift in
the vertical curve formed by the blocks, explained the greater width of the
western side by a falling apart of the blocks. Then, with perfect logic, he
assumed that, if this phenomenon has taken place on the western side, it must
in some measure have taken place on the eastern side as well. Since there the
blocks seem to be perfectly joined, he searched here and there for fissures
and measured them with the micrometer, arriving at a total of 0.006 English
feet or 1.83 mm. Hence, he deducted this rather trivial amount from the length
of the eastern side and assumed that the western side had originally the same
length as the eastern side he so reduced.

In truth, the western side was wider. Penrose himself should have arrived at
this conclusion when he noticed that the western side, and in particular the
southwest corner, had been made substantially higher. Balanos repeated with
greater precision the tests conducted by Penrose. Like Penrose, he tested these
differences in the higher step of the podium, that is, the step below the stylobate,
because the outer edges of the stylobate are too damaged for a precise calculation.
He found that the two ends of the eastern front are on a level and the southwest
corner is raised by 56 mm. This indicates that the southwest corner was intended
to be higher by 10/48 of foot, or 57.8 mm.

There cannot be any doubt that these differences are intentional, contrary
to the opinion of those who follow Ziller in claiming that the differences in
level are proof of a primitive technology. The proof that the differences of
level are intentional is provided by the fact that when the upwards curvature
is measured one arrives at symmetric results for the two halves of each curve
only if the curvature is measured in relation to the lines joining the corners
and not in relation to the level line. Of course, there are scholars, such as
Burn, who deny that these curves were part of the original plan and affirm that
the ground has collapsed in varying amounts at each corner. The Parthenon is
built on solid rock and there has not yet been discovered any evidence of tectonic
movements. To say that the corners have collapsed reveals a mentality similar
to those who argued that the sunspots discovered by Galileo were due to clouding
in the lenses of the telescope. This is proved clearly by the meridian sections
of the Parthenon, which indicate that the floor of the temple rises by deliberate
and regular steps from east to west and from north to south.

Penrose arrived at the following findings for the length of the sides:

North side

69,537.3 mm

South side

69,541.3 mm

Difference

5.0 mm

In my opinion the difference was intended to be 1/48 of foot or 5.78 mm. Later
I will try to explain why I conclude that the intended dimensions were 12,030/48
and 12,031/48 of foot, that is, 69,535.6 and 69,541.4 mm. But Penrose, following
his notion that the Parthenon had to have perfectly square angles, believed
that the dimension of the north side was the original dimension for both sides.

The reason for which the west side and in particular the southwest corner was
made higher is the same for which the west side and the south side were made
slightly longer. This reason appears obvious to anyone who approaches the Parthenon
with his eyes and mind free from preconceived notions: the natural ground is
much lower on the east side and in particular in correspondence with the southwest
corner. From this direction the Parthenon is not seen on a level, but from below.
Vitruvius dedicates an entire chapter (VI.2) to explain that after the proportions
of a building have been calculated, these have to be modified by diminutions
and additions in order to take into account the nature of the site, since it
is not so important that the temple have given proportions, but that it appear
to have them.

As I have already related, Penrose arrived at the following dimensions of the
stylobate, measured in a straight line:

East side

30,889.7 mm

West side

30,896.4 mm

South side

69,541.3 mm

North side

69,537.3 mm

He took the present length of the north side as the more correct one. He assumed
that the present length of the east side is substantially the original one,
except for an infinitesimal excess of 2 mm. due to the coming apart of the blocks
at the joints. Penrose would be the first to agree that measurment is the most
powerful instrument granted to man for the discovery of truth, just because
it brings into clear light any fault of reasoning. Now, his preposterous concern
with an accuracy of 0.006 English feet indicates that he was trying to make
up for a fallacy in approach. He had missed the fact that the stylobate should
have been measured following the line of curvature. He noticed that the normal
intercolumnia seem to be a trifle shorter on the flanks than on the fronts;
but he did not realize that this results from the circumstance that the intervals
projected on a straight line are shorter for the flanks, in which the curvature
is greater.

The difference between straight and curved lines does not affect in a signficant
way the distance between the edge of the stylobate and the axis of the corner
columns. This is the reason why Penrose felt free to formulate a positive statement
about these intervals, reporting that the distance between the axis of the corner
columns and the stylobate is 1023.2 mm on the fronts (I have calculated it as
1023.1 mm.); but on the other intervals, for which he coined the term ”intercolumnial,”
he remained more vague.

The only other investigator who has contributed positively to our information
about the dimensions of the Parthenon is Balanos. His report is honest and competent
within the purpose it sets itself, but he was a curator, not a scholar concerned
with problems of Greek architecture. For this reason his report merely whets
the appetite for information that is missing.

Balanos measured the Parthenon not along the line of the stylobate, but according
to the meridian section, taking the position of the statute of the Maiden as
the center. He arrived at the following results:

Width

30,893 mm.

Length

69,565 mm.

Although he measured the length at the middle of the fronts and the width at
about 1/3 of the length of the flanks, he set the terminal points on the line
joining the corners. His results, as one would expect, are less than my estimate
for the curved lines, 42.6 for the width and 74.7 for the length.

Balanos reported on the length of the intercolumnia one by one. Although he
does not say so, it would seem that they were measured as projected on the line
joining the corners. His figures are the following:

East side

West side

North side

South side

(starting from the north, in mm.)

1,019

1,020

1,020

1,016

3,696

3,668

3,710

3,680

4,290

4,295

4,263

4,294

4,295

4,299

3,693

3,674

4,299

4,295

1,023

1,050

4,290

4,292

69,512

69,521

4,300

4,295

3,662

3,696

1,019

1,020

30,870

30,880

Balanos did not explain the method used in arriving at these results, although
one question at least comes readily to the mind of the attentive reader, since
the addition of the intervals comes to substantially less than the length of
the sides, as estimated by Balanos himself. But the method of measurement by
itself cannot explain entirely the striking irregularities.

From the scanty data that one can glean from the extensive literature on the
columns of the Parthenon, I have come to formulate the following working hypothesis
for explaining in part the irregularity in the intercolumnia. The columns were
the architectural element more difficult to build (we know that they were the
most expensive item on the budget), and they did not come all exactly of the
intended size. Since some were thicker and some were thinner, the builders tried
to compensate by varying the space between the columns. The more perfect columns
apparently were saved for the fronts, with the result that there the intercolumnia
are more regular. This hypothesis could be tested if we had a survey of the
dimensions of all surviving columns taken one by one.

According to Balanos’ figures, there is a great discrepancy in the distances
between the axes of the corner columns and the edge of the stylobate. From his
report we get the following data:

Minimum

1,016 mm

Maximum

1,050

Mean

1,023.4

Median

1,020

Mode

1,022.3

Penrose arrived at the figure of 1023.2 mm. for the distance between the axis
of the corner columns and the front of the stylobate. My figure is 1,023.1 mm.

According to Balanos, a similar sharp discrepancy occurs in the corner intercolumnia:

Minimum

3,662 mm.

Maximum

3,710 mm.

Mean

3,683.6 mm.

Median

3,681.5 mm.

Mode

3,682.5 mm.

My figure is 3,693.5.

6. Before summing up this detailed analysis of the dimensions of the Parthenon
and drawing the general conclusions, it is necessary to consider the height
of the columns.

Balanos reports that the columns have an average (meaning arithmetic mean)
height of 10,433 mm. But, in dealing with the spacing of the columns, I have
reminded the reader of the scientific principle that by quoting averages, without
having first considered the distribution of the items, one arrives at misleading
results. The height of each individual column is not reported, except for the
corner columns. From Balanos we learn that the SE corner column is 10,436 mm.
high, whereas the other three columns have a height of 10,430 mm. This indicates
that the difference was intentional and was calculated as 1/48 of a foot. If
we take 3/20 of the length of the Parthenon, which is 12,030/48 of foot, we
obtain the height of the column, which is 1,804.5/48 or 10,430.3 mm. This is
the height of three of the corner columns; the SE corner column is 1/48 of foot
more, or 10,436.1 mm.

There is a peculiar symmetry in the refinements of the Parthenon, by which
each of the three dimensions of the peripteros was increased by 1/48 of foot
on one side. The length was increased by 1/48 on the south side, the width was
increased by 1/48 on the west side, and the height was increased by 1/48 at
the SE corner.

The height of the columns was related to the length of the peripteros; but
since the length was related to the width, the height must also have been related
to the width.

It appears that tha architect planned the general dimensions so that they could
be expressed in sexagesimal units. Counting in 48th of foot, the length of the
temple was planned as 12,000, its width as 5400, and the height of the columns
as 1800. Most likely these figures were preferred because it was necessary to
calculate the two curvatures of the horizontal lines of the temple and combine
them with a slant outwards of the vertical lines, of which the inclination of
the columns is the main element. As we gather from Vitruvius (III.5.13) the
columns must be made to lean outwards in order to conform to the arcus visionis
of an observer standing in front of the temple. Sexagesimal reckoning is fitting
since angular measurements are involved.

If we change the mentioned figures to 16th of foot, the relations among the
parts appear more obvious: length 400; width 1800; height of columns 600. Hence,
the architect began with proprtions 20:9:3. In feet the dimensions would be
37½ for the height of the columns, 112½ for the width and 250 for
the length.

These are the proportions with which the architect began, but then he introduced
small adjustments by which the width was decreased by 1 foot to 111½ and
conversely the length was increased by 1 foot to 251 feet. The height of the
columns was adjusted so as to keep the proportion 3:20 with the length, since
one could not keep the original proportion both with the length and the width.

The width was decreased by one foot because it was decided that the temple
should be hekatompedos, that is, have a width of 100 geographic feet, which
are equal to 111 1/9 trimmed lesser feet. A width of 111½
feet, which when measured in a straight line is still 111 1/3
feet, comes close enough to the width of 111 1/9 trimmed
lesser feet which would make the Parthenon exactly hekatompedos in its
width.

In planning the width the architect had to take another factor into account.
According to Vitruvius a paramount element in the planning of a temple is the
spacing of the columns on the front. He distinguished (III.3, 1-5) five classes
of temples, of which the one with the columns most close to each other is called
”thick-columned” (pycnostylus) ”in the intercolumniations of which the
thickness of a column and a half can be interposed.” The columns of the Parthenon
were closer to each other even more than in this class of Vitruvius, since they
were intended to occupy half the the width of the front. The columns of the
Parthenon have a diameter of 100 fingers of geographic foot, but the diameter
of the corner columns is incrased to 102 fingers, according to a rule mentioned
by Vitruvius (III.3.11). Hence, the 8 columns of the front occupy a space of
804/16 of geographic foot. One must expect a width of the temple of 1608/16
or 100½ geographic feet. But this width was somehow reduced, since the
temple had to be hekatompedos; the architect settled of 100
1/3 geographic feet, when measuring in a straight line.

The architect also had to achieve dimensions that could be divided into intercolumnia
expressed in round figures. A length of 251 trimmed lesser feet and a width
of 111½ such feet could be divided into intercolumnia of 15½ feet
with corner spaces of 17 feet. As a result of these adjustments the height of
the columns, originally intended as 37½ feet or 1800/48 became 1804.5/48
of foot. The figure was odd, but had the virtue of being 3/20 of the length
measured in a straight line.

In conclusion, the architect began by planning the temple as having a relation
of 9:20 between width and length. Because of a small addition to the length
and a small detraction from the width, the proportion became close to 4:9; the
length is 9/4 of the width, except for an excess of 6/48 or 1/8 of foot. Since
this fact was noticed by Stuart and Revett, it must be asked whether the architect
was conscious of the existence of this proportion. It appears that he was aware
of it, since the decrease of the sides because of curvature is 8/48 of foot
on the front and 18/48 on the flank, with a proportion of 4:9. But this proportion
was not the one that determined his basic plan.

Finally, we must ask what determined the choice of the proportions 9:20 between
width and length. The answer lies in the fact that the Old Temple in its older
part had a width of 50 and a length of 125 trimmed lesser feet. It was decided
to make the Parthenon twice as large, which would make it 100 x 250. But then
the figure of 100 for the width was changed from 100 trimmed lesser feet to
100 geographic feet, since the peripteros of the Old Temple was reckoned in
geographic feet. Hence, the width of the Parthenon, if it is to be hekatompedos
in terms of geographic feet, must be 111.111 trimmed lesser feet. This width
suggests the proportion of 9:20, since the width of 50 feet in the Old Temple
relates as 9:20 to the width of 111.111 feet chosen for the Parthenon. In other
words, we could say that a calculation involving a relation 9:10 was inevitable,
since the Parthenon, like the Old Temple, had to be measured both in trimmed
lesser feet, two modules that relate as 9:10.

According to Vitruvius (III.4.8; IV.4.1) the length of temples should be twice
their width. Nevertheless, it would be difficult to find examples of this proportions
in temples of the classical period. But a clear example is provided by the peripteros
of the Old Temple, which had dimenions of 70 x 140 geographic feet. Hence, it
could be said that the Parthenon had a length equal to twice its width, except
that the width is altered by the relation 9:10, so that the width and length
come to relate as 9:20. A similar shift according to the relation 9:10 occurs
in the columns. The columns have a diameter of 100/16 of geographic foot, or
111.111/16 of trimmed lesser foot, which make clear that 8 columns occupy a
total space equal to half of a width of 100 geographic feet or 111.111 trimmed
lesser feet. The height of the columns should be 600/16 of geographic foot according
to the relation 1:6 between diameter and height, which is the norm for Doric
columns according to Vitruvius (IV.3.4.). But in measuring the height of the
columns of the Parthenon the architect shifted from geographic feet to trimmed
lesser feet, according to the relation 9:10, so that the columns have a height
of 600/16 of foot, but of trimmed lesser foot, having a height of only 9/10
x 600/16 = 540/16 of geographic foot.

Because for political reasons the Parthenon had to be presented as much as
possible as a replacement for the Old Temple, the architect was forced to adopt
unusual proportions between width and length and neglect the usual proportion
based on the near-square 20:21. But it seems that this proportion was too much
a part of the architecture of Greek temples to be overlooked. In the case of
the Parthenon it was brought into play in a more subtle way.

We have seen that the proportions of the Parthenon relate the dimensions of
the stylobate with the height of the columns. Hence, instead of the usual two-dimensional
reckoning of the stylobate of the peripteros, we must be dealing with a three-dimensional
reckoning of the peripteros. The initial plan of the Parthenon assumes a parallepiped
of 112½ x 250 x 37½ feet, so that the peripteros has a volume of 1,054,687.5
cubic feet. From this it follows that in the case of the Parthenon we are not
dealing with a near-square of the type 20:21, but with a near cube of the same
type: a near-cube with sides of 100, 100, 105 feet and a volume of 1,050,000
cubic feet.

If we take the dimensions of the temple as they come to be after the initial
measurements were slightly adjusted, we can reckon 111½ x 251 x 1804.5/48
feet = 1,049,754.9 cubic feet. We can calculate even more precisely by the final
dimensions expressed in 48th of foot:

East side

5344

West side

5345

North side

12,030

South side

12,031

Regular corner columns

1804.5

SE corner column

1805.5

The volume of the peripteros comes to 116,040,023,634.4 cubes with sides of
1/48 of foot. If maximum precision were aimed at in this reckoning, this figure
should be increased by taking into account the slant outwards of the columns.
But there is no need to enter into such details, since it is clear that the
volume of the peripteros was intended to be as close as possible to that of
a near-cube with sides of 100, 100, 105 feet and a volume of 1,050,000 cubic
feet (=116,121,600,000 cubes with sides of 1/48 of foot).

The written texts, both literary and official inscriptions, emphasize that
the Parthenon was hekatompedos in size. Modern scholars project on the ancient
Greeks their own casualness in the matter of measurements, when they claim that
the Parthenon was so called simply because the Athenians had given such a name
to the Old Temple and carried it over to the Parthenon as a matter of habit.
The matter of the size of the Parthenon was far from being a trivial matter
and for this reason the temple was made hekatompedos in three senses:

a) Its width was 100 geographic feet. This was pointed out by Stuart and
Revett.

b) The inner temple, or naos, had a length of 100 trimmed lesser feet.
Penrose gathered from the texts that the naos must have a length of 100 feet,
but was not able to identify properly the part called naos, because he was
thinking in terms of Christian architecture.

c) Finally, my study of the importance of near-squares and near-cubes
in ancient mathematics has led me to realize that the peripteros had a volume
of a near-cube with a basic edge of 100 trimmed lesser cubits. It can be said
that those scholars who tried to explain the term hekatompedos as referring
to the surface of a square with sides of 100 feet, were moving in the right
direction. It is only in terms of its volume that the Parthenon as a whole
can be said to be truly a hekatompedos temple.

7. Once the dimensions of the three Parthenons have been established, it is
possible to explain the reasons for the shifts in plan.

Since Parthenon I had a length of 250 feet like Parthenon III, it is likely
that it, too, had 17 columns on the flanks. The intercolumnia must have been
15½ feet with corner spaces of 17 feet (total 251 feet), as in Parthenon III.

Since Parthenon I had a width of 100 feet, which in principle was that of Parthenon
III, the number of the columns on the fronts must have been 8 as in Parthenon
III. It may be supposed that the intercolumnia on the fronts were 13¾ feet
with corner spaces of 15½ feet or 15¾ feet (total 99¾ or 100¼
feet).

Since the columns were too close to each other on the fronts of Parthenon I,
this drawback could be overcome in Parthenon II, either by making the temple
wider or by reducing the number of the columns. The number of the columns was
reduced because Parthenon II had to be smaller.

Neither Hill nor anybody else has raised the question why Parthenon II was
made smaller, but the explanation is obvious. Parthenon I was intended to be
of poros limestone, whereas Parthenon II was of Pentelic marble. Poros limestone
was a more economical material, because it is spongy and so easty to cut and
to transport, but for the same reason it is a poor material for foundations
since it crumbles under pressure. Hence, the substructure of poros limestone
of Parthenon I could not support a temple of Pentelic marble of the same size
as Parthenon I. Parthenon II was made sharter and narrower so as to reduce the
pressure on the outer edges of the substructure. But the new temple had to be
hekatompedos and had to double the size of the Old Temple: hence the formula
for calculating Parthenon II was based on the dimensions of the Old Tmple C,
instead of being based on the Old Temple B. As a result Parthenon II came to
have dimensions of 84 x 240 feet. The module was shifted to the trimmed lesser
foot, which is the shortest of the modules.

I am inclined to believe that in Parthenon II the number of the columns of
the flanks remained 17, but Hill claims that it was 16. The arguments of Hill
are so obscure, at least as far as I am concerned, as to be beyond comprehension.
He declares:

The dimensions of the stylobate would be 23.510 m. by 66.888 m. This allows
six columns at the ends, with an axial spacing of 4.53 m., and sixteen on the
sides, with an axial spacing of 4.40 m., an arrangement that conforms perfectly
to the standard of the time in which we may be sure this part of the temple
was built. The stylobate blocks for the ends of the temple show a width of 2.09
m., those from the sides a width of 2.04 m., a difference which corresponds
with the difference in the axial spacing on the sides and ends, and indicates
the usual slight variation in the size of the columns.

I do not know what is ”the usual slight variation int he size of the columns.”
In Parthenon III the columns have the same diameter on the fronts and on the
flanks. Hill claims, probably correctly, that the diameter of the columns was
the same in Parthenon II and in Parthenon III. In my opinion the figure that
Hill reports for the width of the stylobate blocks is that of their length.
Possibly the blocks were square, with a width equal to their length. I assume
that each intercolumnium consisted of two blocks, with the axis of the columns
placed on the junction of the blocks, since this is the usual arrangement. But
Hill claims that each intercolumnium consisted of three blocks and that each
column was placed on the middle of a block with a slight overlapping of the
circumference over the blocks on each side, but his demonstration of this supposed
peculiarity is too sktechy to be comprehensible or acceptable.

Unless different data are obtained by repeating the survey conducted by Hill,
it is better to assume that the normal intercolumnium of the flanks was 14¾
trimmed lesser feet (4,092.4 mm).

Possibly the number 111.111 influenced
the architects into choosing a length computed by the factor 9, since 1.111
is the inverse of 9. This is a fact of which the ancients were immediately
aware, since they performed division by multiplying by the inverse. This
method of dividing has practical advantages that recommend it in general,
but, further, it was a necessity for the ancients, who computed by the abacus;
today we employ it in dividing by calculating machines.

Second edition, Munich, 1931, p.
250. What this means is difficult to tell, since Judeich takes cover under
the word beziehungsweise, which is one of the trickiest in the German language
and is even more ambiguous than the English rather.

Dinsmoor reports that a careful
test of the east front gave as a result a length of 30,889 mm. I could quote
this as an absolute confirmation of my major theoretical conclusion, but
so doing would be good rhetoric and bad science, since in matters of measurement
Dinsmoor is addicted to making positive assertions that prove contrary to
empirical facts. Furthermore, even if our informant were a reliable one,
an isolated measurement for which it is not explained how it was obtained
is of limited value, because there is no basis for ascertaining its accuracy
and precision.